
where \(\mathcal {PV}_{U,f}\)  is the perverse sheaf (or monodromic mixed Hodge module) of vanishing cycles defined by applying the vanishing cycles functor \(\phi _f\)  of \(f\)  to \(\mathcal {IC}_U\) , the intersection cohomology complex of \(U\)  with its natural mixed Hodge module structure, and then restricting to \(R\) ; and \(Q_{R,U,f,i}\)  is a \(\mathbb {Z}/2\mathbb {Z}\) -principal bundle giving the difference between the orientation of \(U\)  and the canonical orientation given by \(K^{1/2}_{X,s}\) . The numerical Donaldson-Thomas invariant defined in [1]} is then, as expected, the Euler number of the cohomology of this perverse sheaf. We will use the formalism of d-critical structures in the present article, but we shall not use the formalism of derived geometry and shifted symplectic structures.
